László K.
United States Hopatcong New Jersey

Has anyone read or found any information on what the optimal denominations are for currency? I know that most of us are 'used to' having the 1, 5, and 10 units (plus others) represented in our set of currency, but is this the best set of currency denominations?
I guess the question leads to the following: what set of denominations would allow you to use the least amount of currency to get any particular value? For example using the 1510 set of currency, to get a value of 9 you would need 5 units of currency (5,1,1,1,1). If we added a "2" unit of currency, then we would need only 3 units of currency (5,2,2).
Any help on this topic would be greatly appreciated. Thanks!

Valdir Jorge
Canada Montreal Quebec

What about using powers of 2 (1, 2, 4, 8, 16, 32, ...)? This way, for any small number (smaller than the highest number) you would only need one of each value...

!
United States Brooklyn New York

See here for some discussion of this:
http://www.marginalrevolution.com/marginalrevolution/2003/09... This paper has the solutions: http://graal.enslyon.fr/~yrobert/algo/coins1.pdf
There's a trade off between efficiency in terms of least amount of coins needed, and efficiency in terms of ease of calculation.

Chris Okasaki
United States White Plains New York

You need to constrain the problem some more to get a meaningful answer.
For example, if you want to represent any amount in the smallest number of bills, then you could achieve this by having bills in denominations 1,2,3,4,5,.... Then you can make any amount with only a single bill. But that obviously seems unrealistic.
You probably want something like what set of K denominations allows you to make each amount in the range 1..N in the smallest average number of bills. Or instead of the smallest average number of bills, you might want to minimize the maximum number of bills required for any amount in the range 1..N.
Even if you stick with the smallest average number of bills, you might not want to assume that all amounts between 1..N are equally likely, but rather specify some kind of probability distribution (exponential maybe?).

J C Lawrence
United States Campbell California

Ladislaus wrote: Any help on this topic would be greatly appreciated. Thanks!
Different games have different distributions of value clusters and as such the ideal distribution of denominations is also a function of the game being played. For instance the 18XX games will rapidly teach the 6x multiplication tables up to quite large values as money values used in the game cluster heavily there. That said I'm aware of two common distributions, each centered on multipliers of 4 and 5, both of which have one major variant:
1/5/25/100/500/2,500/10,000/50,000/250,000 etc
1/5/20/100/500/2,000/10,000/50,000/200,000 etc
The two primary variants on these schemes are the optional addition of 2 and 10 denominations. My sense is that 5/25/100 tends to be preferred in America perhaps due to the 25 cent coin used in US currency, and 5/20/100 tends to be preferred in the UK and other parts of the commonwealth perhaps due the 20pence coin used in sterling.

Ed
United States Oakland California
Wankel engine

travistdale wrote:
Holy crap, that is so geeky! I love it.

Scott Russell
United States Clarkston Michigan

1,3,10 is the best, but my group has trouble adding 3's.
I keep 10 because that's how we think. 3 works better than 5, though, for using the fewest chips for each number.

Fraser
Australia Melbourne
Back in the days when there were less maps we played every map back to back
Ooh a little higher, now a bit to the left, a little more, a little more, just a bit more. Oooh yes, that's the spot!

I remember doing an exam question in a computer science subject some time ago (something to do with algorithms).
I believe the question was a proof about which currency set was the best according to the greedy algorithm, i.e. which used the minimum number of units (read coins) to reach any given random amount. I think the exam used a number of existing currency sets from around the world.
The one that was the best was 1, 2, 5, 10, 20, 50 (which is why I would really like a poker chips set of six colours, unlike the standard five colour sets).
The difference to the mathematical paper is it assumes a currency set of five items if I scanned it properly (which is probably unlikely) where as our exam question was using real world coin sets.

Ed
United States Oakland California
Wankel engine

Car Talk had a puzzler on this very issue in today's episode. The question was: if you had 1,000 one dollar bills, how would you package them in envelopes so as to allow you create any sum up to $1,000 using the fewest number of envelopes? The answer is $1, $2, $4, $16, $32, $64, $128, and $256 (all powers of 2).

Jake Di Toro
United States Virginia Beach Virginia

ed95005 wrote: Car Talk had a puzzler on this very issue in today's episode. The question was: if you had 1,000 one dollar bills, how would you package them in envelopes so as to allow you create any sum up to $1,000 using the fewest number of envelopes? The answer is $1, $2, $4, $16, $32, $64, $128, and $256 (all powers of 2).
While nice, and most of us are computer geeks so binary miight be feasable for us. This particular excersize assumed limited curencys. In a real game situation of "unlimited" currency better alternatives exist.

Jim O'Neill (Established 1949)
Scotland Motherwell Graduate of Barlinnie
VENI, VIDI, VISA  my reaction on entering my FLGS.
Like a good red wine, I improve with age... and being laid.

Alas, as with most boardgame developers these days, the traditionalists are left in the cold.
We do not care about efficiency
We do not care about ease of use
We are stepped in the old ways
And I am the baby of this particular group having been born in the first half of the last century (just)
So, where is all this leading?
Bring back Pounds, Shillings and Pence!!!

Fraser
Australia Melbourne
Back in the days when there were less maps we played every map back to back
Ooh a little higher, now a bit to the left, a little more, a little more, just a bit more. Oooh yes, that's the spot!

oneilljgf wrote: Bring back Pounds, Shillings and Pence!!!
What no Guineas or Groats? Where's your sense of tradition?


